Find the value of $k$ for which $kx^2 -5x-12 = 0$ has solutions $x=3$ and $ x = -\frac{4}{3}$.
Recall that for an equation of the form $ax^2 + bx + c = 0$, the sum of roots is equal to $-b/a$ and the product of the roots is equal to $c/a$.

So, we can write the set of equations \begin{align*}
3 - \frac{4}{3} &= \frac{5}{k} \\
-4 &= \frac{-12}{k}
\end{align*}

The second equation tells us immediately that $k = \boxed{3}$.